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DYNAMICAL SYSTEMS

Volume 40, Number 7, July 2020 pp. 4341–4378

BILLIARDS ON PYTHAGOREAN TRIPLES AND THEIR MINKOWSKI FUNCTIONS

Giovanni Panti

Department of Mathematics, Computer Science and Physics University of Udine

via delle Scienze 206 33100 Udine, Italy

(Communicated by Jairo Bochi)

Abstract. It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincar´e disk model. Our tables have m ≥ 3 ideal vertices, and are subject to the restric-tion that reflecrestric-tions in the table walls are induced by matrices in the triangle group PSU±1,1Z[i]. The resulting billiard map eB acts on the de Sitter space x2

1+ x22− x23= 1, and has a natural factor B on the unit circle, the pythagorean

triples appearing as the B-preimages of fixed points. We compute the invari-ant densities of these maps, and prove the Lagrange and Galois theorems: A complex number of unit modulus has a preperiodic (purely periodic) B-orbit precisely when it is quadratic (and isolated from its conjugate by a billiard wall) over Q(i).

Each B as above is a (m − 1)-to-1 orientation-reversing covering map of the circle, a property shared by the group character T (z) = z−(m−1). We prove that there exists a homeomorphism Φ, unique up to postcomposition with elements in a dihedral group, that conjugates B with T ; in particular Φ —whose prototype is the classical Minkowski question mark function— establishes a bijection between the set of points of degree ≤ 2 over Q(i) and the torsion subgroup of the circle. We provide an explicit formula for Φ, and prove that Φ is singular and H¨older continuous with exponent log(m − 1) divided by the maximal periodic mean free path in the associated billiard table.

1. Introduction. Rational points in the real projective line P1R involve two in-tegers, a numerator and a denominator; we can enumerate them by reversing the euclidean algorithm or —equivalently— taking inverse branches of continued frac-tion maps. Rafrac-tional points in the unit circle S1involve three integers, the two legs

and the hypotenuse of a pythagorean triangle. As the line and the circle can be mutually parametrized with preservation of rational points, the complexity of the enumeration is the same, and there is a line of work (starting from [6], and running through [4], [11], [3], [33], [15] and references therein) describing how pythagorean triples can be generated by descending trees.

2010 Mathematics Subject Classification. Primary: 37D40, 11J70.

Key words and phrases. Pythagorean triples, Romik map, billiards, Minkowski function, joint spectral radius.

The author is partially supported by the research project SiDiA of the University of Udine.

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Ascending the same trees amounts to iterating continued fraction maps, and in [42] Romik analyzes one such map, relating it to the geodesic flow on the three-punctured sphere. It turns out that Romik’s map can also be seen as the Gauss map of even continued fractions; see [2, §4], [15, §5], [7, §2] for various developments.

Although there is a birational bijection with rational coefficients between the line and the circle, continued fraction maps on the two spaces are not exactly the same thing. Indeed, the rational symmetry group of the projective line is the extended modular group PSL±2 Z, while that of the circle is SO2,1Z, the stabilizer of the Lorentz form inside SL3Z. When embedded in a larger ambient group —say PSL±2 R— they appear as the (2, 3, ∞) and the (2, 4, ∞) extended triangle groups, and neither is a subgroup of the other (of course, they are commensurable).

In this paper we develop continued fraction maps (of the “slow” type, that is with parabolic fixed points) directly on the circle, as factors of billiard maps determined by ideal polygons in the hyperbolic plane. We summarize our main results as follows: • Let D be a polygon in the Poincar´e disk having m ≥ 3 vertices, all at the boundary at infinity S1. Let B : S1→ S1be the map that sends the interval

between two vertices to the union of the remaining intervals via reflection in the corresponding polygon side. Let T be the group character z 7→ z−(m−1). Then B and T are conjugate by an essentially unique homeomorphism Φ, which provides a bijection between the set of points of degree at most 2 over Q(i) and the torsion subgroup of S1. The homeomorphism Φ is singular and H¨older continuous, of exponent log(m − 1) divided by the maximal mean free path (see Definition 10.3) of periodic trajectories in the hyperbolic billiard determined by D.

The route leading to the above statement is somehow long; we offer two justifica-tions.

1. The end result is a flexible and applicable tool. Indeed, the maximal mean free path referred to above equals twice the logarithm of the joint spectral radius of the set Σ of matrices expressing reflections in the billiard walls. When the vertices of D determine a unimodular partition of S1 (an arithmetical

condition explained in §5), this joint spectral radius can often be explicitly computed; see Example10.6.

2. Along that route we encounter fair landscapes.

We describe our route: in §2 we determine finite sets of reflections generating the orthogonal group O2,1Z and its subgroups SO2,1Z and O↑2,1Z, the latter being

the stabilizer of the upper sheet of the hyperboloid x2

1+ x22− x23= −1. Then, as a

warmup, in §3 we review the construction of the Romik map using our formalism. In §4 we provide explicit PSL±2 R-equivariant bijections between the homogeneous space PSL2R/{diagonal matrices}, the de Sitter space x21+ x22− x23= 1, the space

of oriented geodesics in the hyperbolic plane, and that of quadratic forms of dis-criminant 1. These correspondences are known, but since they appear scattered in the literature and some care is required to extend the acting group from the usual PSL2R to the full PSL±2 R, our brief self-contained treatment in Theorem 4.1may

have some value. In §5 we treat unimodular partitions of the circle; a reader not interested in arithmetical issues may safely skip Theorems5.3and5.5.

The preliminaries being over, we introduce in §6our continued fraction maps B as factors of billiard maps eB associated to ideal polygons whose vertices form a unimodular partition of the circle. Reflections in the table walls are expressed by

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elements of PSU±1,1Z[i] —which we naturally take as matrices— in the Poincar´e model, and by matrices in O↑2,1Z in the Klein model. Here the de Sitter space plays a twofold rˆole, as the phase space of eB as well as the space of shrinking intervals, this double nature being reflected in a double action of PSL±2 R; see Remark 5.2. In §7 we use the bijections in §4 to characterize the natural extension and the absolutely continuous invariant measure of B. In §8 we show that the map B and the extended fuchsian group generated by Σ are orbit-equivalent, and prove the following statement, which combines the classical Lagrange and Galois theorems. A complex number of unit modulus is quadratic over Q(i) if and only if its B-orbit is eventually periodic; moreover, if this is the case, then the conjugate point has the reverse period, and the two points are purely periodic precisely when they are separated by a billiard wall.

In §9 we introduce the conjugacy alluded to above. It is a natural conjugacy; indeed, B is an (m − 1)-to-1 orientation-reversing covering map of the circle, a topo-logical property shared by precisely one group character, namely T (z) = z−(m−1).

We thus have a “linearized” version of a continued fraction map, precisely as the tent map on [0, 1] is a linearized version of the Farey map. It turns out (Lemma9.3) that the natural symbolic coding of points via B, as well as the analogous coding via T , characterizes the ternary betweenness relation on the circle. Since the latter relation determines the circle topology, we obtain in Theorem 9.2 that B and T are conjugate by a homeomorphism Φ, unique up to postcomposition with elements of the dihedral group with 2m elements. This homeomorphism is the analogue of the classical Minkowski question mark function [19], [43], [27], which conjugates the Farey map with the tent map. We provide in Theorem 9.4 an explicit expression for Φ analogous to the Denjoy-Salem formula [43, p. 436] for the question mark function, and show in Examples8.4and9.5how the arithmetic properties of B and T are intertwined by Φ. In Theorem10.1we provide an ergodic-theoretic proof of the fact that Φ has zero derivative at Lebesgue-all points.

In the final Theorem10.5we complete the proof of the connection sketched above between the joint spectral radius of Σ and the H¨older exponent of Φ. In all instances we examined the Lagarias-Wang finiteness conjecture ([31], see §10) turned out to be true for Σ, and a maximizing periodic billiard trajectory was easily guessed and verified. It is plausible that the conjecture holds for all billiard tables determined by unimodular partitions of the circle, and we leave this as an interesting open problem.

2. Notation and preliminaries. Since we treat various spaces of matrices, we will distinguish them notationally, by using boldface for 3 × 3 matrices and lightface for 2 × 2 ones. Points in R3are written in boldface and are always column vectors,

although we may write x = (x1, x2, x3) for typographical reasons. We will use

square or round brackets for vectors and matrices, according whether we are in a projective setting (that is, up to multiplication by nonzero scalars) or in a linear-algebra one. Zero entries in matrices are replaced by blank spaces.

Let L =   1 1 −1  

be the matrix of the three-variable Lorentz quadratic form, and let hx, yi = x>Ly be the corresponding symmetric bilinear map. The upper sheet L = {x : hx, xi =

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−1, x3 > 0} of the 2-sheeted hyperboloid hx, xi = −1 is one of the standard

models of the hyperbolic plane, other models being the upper halfplane H = {z ∈ C : im z > 0}, the Klein disk K = {[x1, x2, x3] ∈ P2R : x21 + x22 < x23}, and

the Poincar´e disk D = {z ∈ C : |z| < 1}; we refer the reader to [10] for an enjoyable introduction to hyperbolic geometry. We need explicit bijections between these models, so we introduce a fifth auxiliary model, namely the upper hemisphere J = {x ∈ R3: x2

1+ x22+ x23= 1, x3> 0}, and state a lemma.

Lemma 2.1. The spaces L, H, K, D, J are in bijective correspondence via the commuting diagram

L

K

J

H

D

π τ0 η υ µ τ C where • π : R3\ {0} → P2

R is the natural quotient map, • τ0 is the stereographic projection through (0, 0, −1),

• η(x) = (x1+ i)/(x3− x2),

• υ([x1, x2, x3]) = x1/x3, x2/x3, (x23− x21− x22)1/2/x3 is the “vertical”

projec-tion,

• µ is the stereographic projection through (0, 1, 0) to the halfplane {x2= 0, x3>

0}, followed by the obvious identification of the latter with H, • τ is the stereographic projection through (0, 0, −1) to the disk {x2

1 + x22 <

1, x3= 0}, followed by the obvious identification of the latter with D,

• C is the M¨obius transformation z 7→ C ∗ z = (z − i)/(−iz + 1) induced by the Cayley matrix C = 2−1/2−i 11 −i ∈ PSL2C (as customary, we blur the distinction between matrices and the maps they induce).

These correspondences extend to the respective ideal boundaries.

Proof. The proof reduces to a commentary on the figure on page 70 of [10]. The upper-left triangle commutes because υ◦π sends x = (x1, x2, x3) ∈ L to x1, x2, (x23−

x2

1− x22)1/2/x3 = (x1, x2, 1)/x3 = (1/x3)(x1, x2, x3) + (1 − 1/x3)(0, 0, −1). The

upper-right triangle commutes because µ sends (x1, x2, 1)/x3∈ J to x1/(x3− x2) +

i/(x3− x2) = η(x). The lower-right triangle commutes because, given y ∈ J ,

C ∗ µ(y) = 1 −i −i 1  ∗ y1+ y3i 1 − y2 = y1+ y2i 1 + y3 = τ (y).

The fact that these correspondences extend to the ideal boundaries is obvious as soon as the boundary ∂L of L and the maps π, τ0, η on it are properly defined.

We see ∂L as the intersection of the projective closure of L ∪ (−L) (i.e., the variety x21+ x22− x23+ x24= 0 in P

3

R) with the plane at infinity x4= 0, and we set

π([x1, x2, x3, 0]) = [x1, x2, x3],

τ0([x1, x2, x3, 0]) = (x1/x3, x2/x3, 0),

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We can then view [x1, x2, x3, 0] ∈ ∂L as the limit (in the euclidean metric of an

appropriate local chart) of x(t) = t x1, x2, (x21+ x22+ 1/t2)1/2 ∈ L, for t → +∞.

An easy computation shows that the π-, τ0-, η-images of [x1, x2, x3, 0] ∈ ∂L, as

defined above, agree with the limits (in the euclidean metric) of π(x(t)), τ0(x(t)),

η(x(t)), for t → +∞. This guarantees the required commutativity.

It is well known that the orthogonal group O2,1R of the Lorentz form has four connected components, namely the component of the identity (which is a normal subgroup) and its cosets with respect to the diagonal matrices having diagonal entries (−1, 1, 1), (1, 1, −1), (−1, 1, −1). The union of the component of the iden-tity with its (−1, 1, −1)-coset is the special orthogonal group SO2,1R, while its union with the (−1, 1, 1)-coset is the group O↑2,1R of all matrices that preserve L; equivalently, O↑2,1R = {A ∈ O2,1R : the (3, 3)-entry of A is > 0}. We will write SO↑2,1R = SO2,1R ∩ O↑2,1R for the component of the identity.

The group of isometries (including the orientation-reversing ones) of H is PSL±2 R = {A ∈ GL2R : |det A| = 1}/{±I}, which acts on H as follows: given A = a bc d,

then A ∗ z equals (az + b)/(cz + d) if det A = 1, and equals (a¯z + b)/(c¯z + d) if det A = −1. Conjugating PSL±2 R with the Cayley matrix we obtain the group

PSU±1,1C =α ββ¯ α¯ 

∈ GL2C : |α|2− |β|2 = 1 .

± I, which acts on D via

α β ¯ β α¯  ∗ z = ( (αz + β)/( ¯βz + ¯α), if |α|2− |β|2= 1; (β ¯z + α)/( ¯α¯z + ¯β), if |α|2− |β|2= −1.

We construct an isomorphic representation PSL±2 R → O↑2,1R by identifying the

vector w = (w1, w2, w3) ∈ R3 with the matrix

W =−w2+ w3 −w1 −w1 w2+ w3



, (1)

on which A ∈ PSL±2 R acts on the left by W 7→ (A−1)>W A−1. This is a well defined action, independent from the lift of A to SL±2 R, linear, and preserving the form hw, wi = − det W . Computing the images of the 1-parameter subgroups in the Iwa-sawa decomposition of PSL2R provides a geometric picture of the representation, namely cos(t) − sin(t) sin(t) cos(t)  7→   cos(−2t) − sin(−2t) sin(−2t) cos(−2t) 1  , exp(t/2) exp(−t/2)  7→   1 cosh(t) sinh(t) sinh(t) cosh(t)  , 1 t 1  7→   1 −t t t 1 − t2/2 t2/2 t −t2/2 1 + t2/2  . (2)

Convention 2.2. In order to simplify notation we adopt the convention that, when-ever a matrix in PSL±2 R is denoted by a certain capital letter, then its image under

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the above representation, and its C-conjugate, are denoted by the same capital let-ter in bold and in calligraphic fonts, respectively. With this understanding, we give names to a few matrices that will recur throughout this paper.

J =−1 1  , J =  −i i  , J =   −1 1 1  , F =  1 1  , F =  1 1  , F =   1 −1 1  , P =−1 2 1  , P =  i 1 − i 1 + i −i  , P =   −1 −2 2 −2 −1 2 −2 −2 3  , G =√1 2 1 1 1 −1  , G = √1 2  1 + i 1 − i  , G =   1 1 1  . (3)

Explicit computation —which we omit— shows that η ◦ A = A ◦ η on L, for every A in the above 1-parameter subgroups, and also for A = J ; therefore the identity η ◦ A = A ◦ η holds for every A ∈ PSL±2 R. The action of O↑2,1R on

R3 descends to a projective action on P2R that fixes the Klein model K and its boundary ∂K. These observations, together with Lemma2.1, imply that for every A ∈ PSL±2 R the diagram

K

D

H

K

D

H

τ ◦υ A A C A τ ◦υ C (4)

commutes. The analogous diagram involving the ideal boundaries of K, D, H com-mutes as well, and actually simplifies. Indeed, the nontrivial bijection τ ◦ υ reduces on ∂K to the obvious identification [x1, x2, x3] 7→ (x1+ x2i)/x3, while C−1 ◦ τ ◦

υ reduces to the stereographic projection through [0, 1, 1], namely [x1, x2, x3] 7→

x1/(x3− x2). We will thus switch freely between ∂K and ∂D, using S1 as a neutral

name for both.

Let D be a polygon in H, bounded by m ≥ 3 geodesics l0, . . . , lm−1, and

hav-ing angles at vertices π/e0, . . . , π/em−1, with e0, . . . , em−1integers ≥ 2 or ∞ (if the

corresponding vertex lies in ∂H); the Gauss-Bonnet formula forces m − 2 >P

ae −1 a .

The extended Coxeter group associated to D is the subgroup Γ±of PSL±2 R gener-ated by the reflections in the sides of D. It has the presentation

hx0, . . . , xm−1|x20= · · · = x 2

m−1= (x0x1)e0 = · · · = (xm−1x0)em−1 = 1i

(with the understanding that relators (xaxa+1)∞ do not appear), and D is a

fun-damental domain for it. Its index-2 subgroup of orientation-preserving elements Γ = Γ±∩ PSL2R is a fuchsian group of finite covolume; see [28], [32]. When D is a triangle we write ∆(e0, e1, e2) and ∆±(e0, e1, e2) for Γ and Γ±, referring to them

as a triangle group and an extended triangle group, respectively (the adjective ex-tended stresses the fact that orientation-reversing isometries are allowed; in both cases, the action on H is properly discontinuous). Note that the numbers e0, e1, e2

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will freely use all of the above terminology when working in other models of the hyperbolic plane.

Let us return to the Lorentz form h–, –i. We recall that, given a nonisotropic vector w, the reflection Rw is the unique linear involution of R3 that fixes

point-wise the polar hyperplane {x : hw, xi = 0} and exchanges w with −w. An easy computation (of course, all of this is well known) shows that:

(i)

Rw(x) = x −

2hw, xi

hw, wiw, (5)

(ii) Rw preserves h–, –i,

(iii) in terms of matrices,

Rw= I −

2 hw, wiw w

>L, (6)

(iv) Rw∈ O↑2,1R if and only if hw, wi > 0.

Notation 2.3. • O2,1Z (respectively, SO2,1Z, O↑2,1Z, SO ↑ 2,1Z) is the intersec-tion of O2,1R (respectively, SO2,1R, O↑2,1R, SO ↑ 2,1R) with GL3Z. • PSL±2 Z = {A ∈ PSL±2 R : A has entries in Z}.

• PSU±1,1Z[i] = {A ∈ PSU±1,1C : A has entries in Z[i]}.

• hF, P, Gi+ (and analogously for other groups generated by involutions) is the

group of all products of an even number of elements in {F, P, G}.

The four matrices J , F , P , G in (3) are in O↑2,1Z; in particular they are of the form Rw, for w equal to (1, 0, 0), (0, 1, 0), (1, 1, 1), (−1, 1, 0), respectively. In [35] it

is proved that the five reflections J , F , R(0,0,1)= diag(1, 1, −1), R(1,1,0) = J GJ ,

P generate O2,1Z (see [17] for an elementary proof which avoids the theory of Kac-Moody Lie algebras); we give an independent and expanded version in the following theorem.

Theorem 2.4. We have O↑2,1Z = hF , P , Gi, which is isomorphic to the extended triangle group ∆±(2, 4, ∞); adding R(0,0,1) as a further generator we obtain the full

group O2,1Z. The group hF , P , J i is an index-2 subgroup of O↑2,1Z, and equals

∆±(2, ∞, ∞); its image hF , P, J i inside PSU±1,1C is PSU±1,1Z[i].

Proof. We work in H. Let Γ = {A ∈ PSL2R : A ∈ SO↑2,1Z}; then, by definition, Γ

is an arithmetic fuchsian group. We observe that hF, P, Gi+ is the triangle group ∆(2, 4, ∞). Indeed F, P, G are the reflections in the three geodesics

• l0, whose endpoints are 1 and −1;

• l1, whose endpoints are ∞ and 1;

• l2, whose endpoints are 1 −

2 and 1 +√2.

These geodesics determine a triangle D in H with vertices at 1 + i√2 with angle π/2, at i with angle π/4, and at the ideal point 1 with angle 0.

Clearly hF, P, Gi+ is a subgroup of Γ, and it is well-known that a fuchsian group containing a triangle group must itself be a triangle group [44, §6]. The partially ordered set of all nine non-cocompact arithmetic triangle groups has been deter-mined by Takeuchi in [46], and ∆(2, 4, ∞) is maximal in it; therefore Γ = hF, P, Gi+.

Adding F as a further generator to hF, P, Gi+we obtain hF, P, Gi = {A ∈ PSL±2 R : A ∈ O↑2,1Z}, as claimed.

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For the second statement, observe that replacing the generator G with J means replacing l2 with the geodesic l20 whose endpoints are 0 and ∞. The polygon

de-termined by l0, l1, l02 is the triangle D0 = D ∪ G[D], with angles π/2 at i, and 0

at 1 and at ∞; hence hF, P, J i is the extended triangle group ∆±(2, ∞, ∞). Clearly hF , P, J i ≤ PSU±1,1Z[i], and by computing

C−1a + bi c + di c − di a − bi  C = a + d b + c −b + c a − d  , we see that C−1 PSU±

1,1Z[i]C is a subgroup of PSL ±

2 Z. Taking into account the

respective fundamental domains, it is easy to check that hF, P, J i has index 3 in PSL±2 Z; therefore C−1 PSU±1,1Z[i]C equals either hF, P, Ji or the full PSL±

2 Z.

However, this second possibility is ruled out by the fact that PSL±2 Z (which is the extended (2, 3, ∞) triangle group) contains elements of order 3, and hence of trace 1 (up to sign), while clearly no element of PSU±1,1Z[i] may have trace 1.

3. Pythagorean triples and the Romik map. A [primitive] pythagorean triple is a point t = (t1, t2, t3) ∈ Z3 such that t3 > 0, gcd(t1, t2, t3) = 1, and t21+ t22 =

t23. Pythagorean triples correspond bijectively to rational points in the unit circle,

which in turn correspond, via stereographic projection, to points in P1

Q. These correspondences provide various techniques for enumerating triples, among which the one known to Euclid: given any reduced fraction a/b, the triple (a2−b2, 2ab, a2+

b2)/ gcd(a2−b2, 2ab, a2+b2) is pythagorean, and every pythagorean triple is uniquely

obtainable in this way (the gcd in the denominator is 1 if 2 | ab, and 2 otherwise). As noted in the introduction, many techniques are cast in the form of the descent of a binary or ternary tree.

A remarkable connection with the theory of continued fractions is offered in [42]; as a warmup, we sketch it using our notation. We partition S1 in four quarters

I0, I1, I2, I3, with Ia = {exp(2πti) : a/4 ≤ t ≤ (a + 1)/4}. Let A = R(1,−1,1) =

F P F . Then A acts on S1 (viewed as ∂K, see the diagram (4) and the resulting identifications) by exchanging x with the other point of intersection of S1 with the line through x and [1, −1, 1]; the interval I3 is thus bijectively mapped to

the union of the other three intervals. We fold back I0∪ I1∪ I2 to I3 via the

reflection F acting on I0, the rotation J F on I1, and the reflection J on I2; see

Figure1. Conjugating this process via the stereographic projection through [0, 1, 1] we obtain the Romik map in Figure2. By construction, it is a continuous piecewise-projective selfmap of the real unit interval [0, 1]. It is composed of three pieces, each one mapping bijectively a subinterval of [0, 1] to the whole interval. The computation of these pieces is built-in in our formalism: indeed, since stereographic projection from [0, 1, 1] is C−1◦ τ ◦ υ on ∂K, computation amounts to switching from boldface to lightface. Thus, the first piece is induced by J (F P F ) =  1

−2 1



acting on F P F J ∗ [0, 1] = [0, 1/3], the second one by (J F )(F P F ) = J P F =−2 1

1



acting on F P J ∗ [0, 1] = [1/3, 1/2], and the third by F (F P F ) = P F =2 −1 1  on

F P ∗ [0, 1] = [1/2, 1].

We adopt another notational shorthand, by consistently writing t, θ (or s, σ, . . .) for pairs t = [t1, t2, t3] ∈ ∂K, θ = (t1+ t2i)/t3 ∈ ∂D, identified as in the discussion

following the diagram (4). We recall that the residue field of the point t = [t1, t2, t3]

in the projective variety {x21+ x22− x23 = 0} = ∂K is Q(t) = Q(t1/t3, t2/t3). If

Q(t) = Q we say that t is a rational point; in this case t has a canonical presentation as a pythagorean triple. The corresponding θ ∈ Q(i) has a canonical presentation

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[0, −1, 1] [1, 0, 1] [0, 1, 1] [−1, 0, 1] A[0, 1, 1] = [4, −3, 5] [1, −1, 1]

Figure 1. A hint of the construction of the Romik map; the in-terval I3 and its stereographic projection to [0, 1] as thick lines

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Figure 2. The Romik map.

as well, but a subtler one. For each prime integer p ≡ 1 (mod 4), write uniquely p = a2+ b2, for integers a > b > 0, and let θp = (a + bi)/(a − bi) (corresponding,

as in Euclid’s setting, to tp = [a2− b2, 2ab, a2+ b2]). It is well known —and easy

to prove [20]— that every θ ∈ S1∩ Q(i) factors uniquely in Q(i) as a product of a unit in Z[i] and finitely many numbers θp and their inverses. This implies that the

set of primitive pythagorean triples forms a multiplicative group, isomorphic to the direct sum of the cyclic group of order 4 with countably many copies of the infinite cyclic group. We thus obtain our second canonical presentation: every θ ∈ S1

∩Q(i) can be uniquely expressed as θ = κµ/¯µ, with κ ∈ {1, i, −1, −i} and µ ∈ Z[i] having prime decomposition of the form

µ = (a1+ b1i)e1· · · (aq+ bqi)eq,

with aj > |bj| > 0, ej > 0 for every j, and the pairs (a1, |b1|), . . . , (aq, |bq|) all

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4. The de Sitter space. The de Sitter space is the one-sheeted hyperboloid S = {x ∈ R3: hx, xi = 1}; it is a lorentzian manifold of constant positive curvature [37],

[36]. The de Sitter space is in natural bijection with various spaces of interest to us: these bijections are well known, albeit a bit scattered in the literature. We collect the relevant facts in Theorem4.1, whose nonstandard feature is the rˆole of PSL±2 R as the acting group, instead of the usual PSL2R.

We recall from §2 that A 7→ A is a group isomorphism from PSL±2 R to O2,1R. We define now another isomorphism Λ : PSL±2 R → SO2,1R by Λ(A) = (det A)A. In the following theorem we let e : {1, −1} → {0, 1} have value 0 on 1, and 1 on −1; also, we denote any group action by a star.

Theorem 4.1. The spaces in the following list, together with the specified base points and transitive left actions of PSL±2 R, are in bijective correspondence. These

correspondences preserve the base points and are equivariant with respect to the actions.

(S1) The de Sitter space S, with base point (1, 0, 0) and action A ∗ x = Λ(A)x. (S2) The coset space PSL2R/A, for A the subgroup of diagonal matrices, with base

point A and action A ∗ EA = AEJe(det A)A.

(S3) (P1R × P1R) \ (diagonal), with base point (∞, 0) and action A ∗ (ω, α) = (A ∗ ω, A ∗ α).

(S4) (S1× S1) \ (diagonal), with base point (i, −i) and action A ∗ (σ, ρ) = (A ∗

σ,A ∗ ρ).

(S5) The space of oriented geodesics in D, with base point the geodesic from −i to i and action A ∗ g =A [g].

(S6) The space of quadratic forms q xy = q1x2− q2xy + q3y2 of discriminant 1,

with base point −xy and action (A ∗ q) xy = (det A)q A−1 xy.

Each space carries a PSL±2 R-invariant infinite measure, which is the quotient Haar measure in (S2), and is induced by the form (ω − α)−2dω dα in (S3). In (S1), the measure of a Borel subset B of S is the euclidean volume of the cone {tx : t ∈ [0, 1], x ∈ B}, and analogously for (S6).

Proof. The natural bijections among the spaces in (S3), (S4), (S5) are the obvious ones resulting from the diagram (4). Here we will first describe the bijections among (S2), (S3), (S6), and then the one between (S1) and (S6).

Let q be a form as in (S6), associated to the symmetric matrix

Q =

q1 −q2/2

−q2/2 q3

!

, (7)

of determinant −1/4. We obtain a pair (ω, α) as in (S3) by labeling the two roots of q(x, 1) as follows:

(a) if q1= 0 and q2= 1, then ω = ∞ and α = q3;

(b) if q1= 0 and q2= −1, then ω = −q3 and α = ∞;

(c) if q16= 0, then ω = q2+ 1 2q1 , α = q2− 1 2q1 .

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Given a pair (ω, α) as in (S3), we set E =                      " 1 α 1 # , if ω = ∞; " ω −1 1 # , if α = ∞; |ω − α|−1/2 " ω α 1 1 # Je(sgn(ω−α)), otherwise;

thus defining a coset EA as in (S2).

Finally, any EA in (S2) determines a symmetric matrix Q0 of determinant −1/4 via Q0 = −1 2(E −1)>  1 1  E−1;

note that Q0 is well defined, i.e., independent from the choice of a representative in

EA and from the lift of this representative to SL2R.

It is clear that each of these constructions preserves the base points and is equi-variant with respect to the listed actions. Therefore, the claimed correspondence between (S2), (S3), (S6) follows as soon as we prove that the final Q0 equals the starting Q. We check case (c), leaving the simpler cases (a) and (b) to the reader. By definition, E = 1 2|q1|1/2 q2+ 1 q2− 1 2q1 2q1  Je(sgn q1), so that E−1= 1 2|q1|1/2 Je(sgn q1) 2q1 −q2+ 1 −2q1 q2+ 1  . Hence Q0= − 1 8|q1|  2q1 −2q1 −q2+ 1 q2+ 1  Je(sgn q1)  1 1  Je(sgn q1) 2q1 −q2+ 1 −2q1 q2+ 1  = −(sgn q1) 1 8|q1|  2q1 −2q1 −q2+ 1 q2+ 1   1 1   2q1 −q2+ 1 −2q1 q2+ 1  (8) = − 1 8q1 −8q2 1 4q1q2 4q1q2 −2q22+ 2  ,

which is the initial Q; note the use of the identity J±1 1

1 J±1 = ±1 11 in the

computation.

The bijection between (S1) and (S6) is a simple change of variables, namely   w1 w2 w3  =   1 −1 1 1 1     q1 q2 q3  . (9)

This change of variables transforms the matrix Q in (7) to W/2, where W is the matrix in (1). This implies that the bijection is equivariant with respect to the actions listed in (S1) and (S6); see also Remark5.2.

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For future reference we list here the form q and the point w ∈ S as a function of (ω, α): q = −xy + αy2, w = (1, α, α), if ω = ∞; q = xy − ωy2, w = −(1, ω, ω), if α = ∞; (10) q = x 2− (ω + α)xy + ωαy2 ω − α , w = (ω + α, ωα − 1, ωα + 1) ω − α , otherwise.

5. Circle intervals. The unit circle S1 is cyclically ordered by the ternary

be-tweenness relation t ≺ x ≺ t0, which reads “t, t0, x are pairwise distinct, and trav-eling from t to t0 counterclockwise we meet x”. Every pair of distinct points t, t0 determines two closed intervals, namely [t, t0] = {t, t0} ∪ {x : t ≺ x ≺ t0} and

[t0, t]. Given w in the de Sitter space, the set Iw = {x ∈ S1: x3hw, xi ≥ 0} is an

interval as well (the factor x3, i.e., the third coordinate of x, makes the definition

independent from the choice of a representative for x). Let us denote the ordinary cross product of two vectors in R3 by x × y.

Lemma 5.1. Let t, t0∈ S1 be distinct, and let

w = Lt

0× Lt

ht0, ti , (11)

the right-hand side being independent from the chosen lifts of t, t0 to R3\ {0}. Then

the following statements hold. (i) w ∈ S, and Iw= [t, t0].

(ii) Let (ω, α) ∈ (P1R×P1R)\(diagonal) be the pair corresponding to w according to Theorem4.1. Then we have

(ω, α) = (µ ◦ υ)(t0), (µ ◦ υ)(t).

(iii) For every A ∈ O↑2,1R, we have A[Iw] = IAw, which equals [At, At0] if

det A = 1, and [At0, At] otherwise.

(iv) w ∈ Q3 if and only if both t and t0 are rational points.

(v) The arclength of [t, t0] and the third coordinate w3 of w are related by

arclength([t, t0]) = 2 arccot(w3).

(vi) If t and t0 do not lie on the same diameter (i.e., by (v), if w36= 0), then the

unique circle in R2 perpendicular to S1 and passing through t, t0 has center

(w1/w3, w2/w3) and curvature |w3|.

(vii) Assume that

Iw0⊇ Iw1⊇ Iw2⊇ · · · ,

with arclength tending to 0 (i.e., limt→∞wt,3= ∞). Then limt→∞arclength

(Iwt)(2/wt,3) = 1. Proof. (i) Every rotation

S =   cos s − sin s sin s cos s 1  

leaves invariant the arclength of [t, t0] and the third coordinate of w (because S belongs to SO3R as well as to SO2,1R, and hence (LSt0× LSt)hSt0, Sti = Sw). Therefore we assume without loss of generality t = [1, 0, 1] and t0 = [cos r, sin r, 1], for some 0 < r < 2π. Then, by explicit computation, w = (sin r)(1 − cos r), 1, (sin r)(1 − cos r), which is indeed in S. Let x(u) = [cos u, sin u, 1], and let f (u) =

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hw, x(u)i : [0, 2π) → R. Then, by elementary projective geometry, f takes value 0 in precisely two points, namely in u = 0 and in the unique solution to x(u) = t0. Again by explicit computation, f has derivative f0(u) = cos u − (sin r)(sin u)/(1 − cos r), which is positive at 0. This, and extending f to be periodic, then implies that hw, xi ≥ 0 if and only if x ∈ [t, t0], as claimed.

(ii) We have (µ ◦ υ)−1(ω) = (2ω, ω2− 1, ω2 + 1), and analogously for α. Our

statement amounts then to the verification that the vector L(2ω, ω2− 1, ω2+ 1) × L(2α, α2− 1, α2+ 1)

h(2ω, ω2− 1, ω2+ 1), (2α, α2− 1, α2+ 1)i

resulting from (11) equals the vector w given by (10). This is a straightforward computation.

(iii) Let x be a point in S1, and choose a representative for it with positive third

coordinate. Then, for every A ∈ O↑2,1R, the third coordinate of A−1x is still positive; we thus have x ∈ A[Iw] iff A−1x ∈ Iw iff hw, A−1xi ≥ 0 iff hAw, xi ≥ 0

iff x ∈ IAw. The second statement follows from the first and the remark that

t ≺ A−1x ≺ t0 is equivalent to At ≺ x ≺ At0 if det A = 1, and to At0 ≺ x ≺ At if det A = −1.

(iv) The right-to-left implication follows from the definition of w. Conversely, if w ∈ Q3 then the proof of the equivalence between (S1) and (S6) in Theorem 4.1

yields that the form q corresponding to w has rational coefficients. Since q has discriminant 1, the roots of q(x, 1) (given by (a), (b), (c) in the proof of the same Theorem4.1) are rational numbers. By (ii), t and t0 are the reverse stereographic projections through [0, 1, 1] of these roots, and thus are rational points.

(v) As in (i), we assume t = [1, 0, 1] and t0 = [cos r, sin r, 1]. Then, as computed in (i), w3= (sin r)(1 − cos r) = cot(r/2), and our statement follows.

(vi) Looking at w as a point in P2R, the identities hw, ti = hw, t0i = 0 mean that w

is the intersection point of the two lines tangent to S1at t and t0; thus the described

circle has center (w1/w3, w2/w3). Upon applying the rotation in the proof of (i),

the statement about the curvature follows by direct inspection. (vii) This is clear.

Remark 5.2. Since, as it is easily seen, the map w 7→ Iw is a bijection between

S and the space of closed circle intervals, it is tempting to add a seventh item to the list in Theorem 4.1. However this would not be correct, since the action in Lemma 5.1(iii) does not agree with the one in Theorem 4.1(S1). In other words, PSL±2 R acts on the space of intervals via the “bold” isomorphism A 7→ A, while it acts on the de Sitter space via Λ. The following commuting diagram may clarify the situation O↑2,1R O2,1R PSU±1,1C PSL±2 R SO2,1R O2,1R C−1– C Λ bold (12)

In (12), the rightmost vertical arrow is the involutive automorphism A 7→ (det A) (sgn A3,3)A of O2,1R, which restricts to the isomorphisms Λ◦bold−1and bold ◦Λ−1.

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Since these isomorphisms obviously preserve the fact that a matrix has integer entries, Theorem 2.4 implies that SO2,1Z = ΛhF, P, Gi = h−F , −P , −Gi ' ∆±(2, 4, ∞) and SO↑2,1Z = ΛhF, P, Gi+ = hF , P , Gi+' ∆(2, 4, ∞).

When working with continued fractions algorithms one naturally deals with uni-modular intervals in P1R, namely intervals [p/q, p0/q0] with rational endpoints and such that det p pq q00 = −1; for example, the intervals [1/(a + 1), 1/a] of continuity for the Gauss map x 7→ 1/x − b1/xc are unimodular. It is a trivial —but key— fact that the modular group PSL2Z acts simply transitively on such intervals. The situation for intervals on the circle is more involved.

Theorem 5.3. The set S ∩ Z3 is partitioned in two orbits, corresponding to the

parity of w3, by the action of SO↑2,1Z. On each orbit the action is simply transitive.

Replacing SO↑2,1Z with its index-2 subgroup ΛhF, P, Ji+ each orbit is further split

in two.

Proof. It is easy to check that each of −F , −P , −G preserves the parity of w3;

hence there are at least two orbits.

Choose w ∈ S ∩ Z3 and let (ω, α) ∈ (P1

Q × P1Q) \ (diagonal) be the corre-sponding ordered pair according to Theorem 4.1. An appropriate power (F P )k

of the parabolic matrix F P (that fixes 1) sends (ω, α) to a new pair (ω0, α0) with 0 ≤ ω0 ≤ 1. By [42, Theorem 2(i)], the orbit ω0 = ω00, ω10, ω20, . . . of ω0 under the Romik map ends up after finitely many steps, say the nth step, in one of the two parabolic fixed points 0, 1. For each 0 ≤ t < n, let

At=      J F P F, if 0 < ω0t< 1/3; J P F, if 1/3 ≤ ω0t< 1/2; P F, if 1/2 ≤ ω0t< 1;

be the matrix acting at time t. Then A = F J An−1An−2· · · A0(F P )k ∈ hF, P, J i,

and A∗(ω, α) = (ω00, α00) is such that ω00∈ {∞, −1}. Postcomposing A, if necessary, with J (if ω00= ∞) or with F (if ω00= −1), we have A ∈ hF, P, J i+.

Suppose ω00= ∞. Then α00∈ Z because the point w00corresponding to (∞, α00) equals (1, α00, α00) by (10), and also equals Λ(A)w, which is a point in Z3. This

implies that an appropriate power of the parabolic matrix P J =1 2

1 maps (∞, α00)

either to (∞, 0) or to (∞, 1). If, on the other hand, ω00 = −1, then the same argument with P J replaced by (J P J )F =  2 1

−1  (which is parabolic fixing −1)

yields that a power of J P J F maps (−1, α00) either to (−1, 1) or to (−1, ∞). Summing up, we have proved that the pair (ω, α) is in the hF, P, J i+-orbit of

one of the pairs (∞, 0), (∞, 1), (−1, 1), (−1, ∞). Now, the rotation GF ∈ hF, P, Gi+

maps the first pair to the third, and the second to the fourth. By Theorem 4.1

this means that the original point w is in the ΛhF, P, Gi+-orbit of either (1, 0, 0)

or of (1, 1, 1). Since ΛhF, P, Gi+

= SO↑2,1Z by Remark 5.2, our first claim is established.

Simple transitivity follows from the fact that both (∞, 0) and (∞, 1) have triv-ial stabilizer in hF, P, Gi+ (because an element of a fuchsian group that fixes two distinct cusps must be the identity).

Finally, the pairs (∞, 0), (∞, 1), (−1, 1), (−1, ∞) remain distinct modulo hF, P, J i+. Indeed, the latter is the triangle group ∆(2, ∞, ∞), which has two distinct cusp orbits, and it is easy to check that any identification of the above four pairs would collapse these two orbits.

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We can now define unimodularity for circle intervals. Definition 5.4. Let t, t0 be distinct rational points in S1

, and let w ∈ S ∩ Q3 be

the point corresponding to [t, t0] according to Lemma5.1. If w ∈ Z3and w3is even

(odd), then we say that [t, t0] is an even (odd) unimodular interval.

Theorem 5.5. Let t, t0, w be as in Definition5.4; then the following conditions are equivalent.

1. [t, t0] is unimodular (either even or odd). 2. Rw has integer entries.

3. [t, t0] is the image either of [0, −1, 1], [0, 1, 1] or of [1, 0, 1], [0, 1, 1] under some (necessarily unique) element of SO↑2,1Z.

4. ht, t0i ∈ {−1, −2} (here t, t0are the canonical presentations of t, t0 as primitive

pythagorean triples).

If these conditions hold, then [t, t0] is odd iff it is the image of [1, 0, 1], [0, 1, 1] iff ht, t0i = −1. Moreover, R

w belongs to hF , P , J i, and the matrixRw∈ PSU±1,1Z[i]

corresponding to it under Convention 2.2is θ θ0 1 1  Jθ θ 0 1 1 −1 , (13) where θ, θ0∈ S1

∩ Q(i) are identified with t, t0 as in §3.

Proof. (1) ⇒ (2) Since hw, wi = 1, this is immediate from the explicit formula for Rw in (6).

(2) ⇒ (3) Let

(ω, α) = (µ ◦ υ)(t0), (µ ◦ υ)(t) ∈ (P1Q × P1Q) \ (diagonal)

(see Lemma 5.1(ii)). Then, as in the proof of Theorem 5.3, we construct A ∈ hF, P, J i+such that A ∗ (ω, α) equals either (∞, α00) or (−1, α00). Since F G ∗ (−1) =

∞, there exists B ∈ hF, P, Gi+

with B ∗ (ω, α) = (∞, q), for some q ∈ Q. Hence, Λ(B)w = (1, q, q) = v. We then have

Λ(B)RwΛ(B)−1 = RΛ(B)w = Rv= I −

2 hv, viv v

>L,

and the leftmost entry in the display is a matrix with integer entries. Multiplying through by −1, subtracting the identity matrix I, and multiplying by L on the right, we see that the matrix

2 hv, vivv > = 2   1 q q q q2 q2 q q2 q2  

must have integer entries. This implies that the denominator of the rational num-ber q must divide 2, and so must do the denominator of q2; therefore q is an integer.

Thus, as in the proof of Theorem5.3, an appropriate power (P J )k will map (1, q, q)

either to (1, 0, 0) or to (1, 1, 1); therefore, Λ (P J )kBw ∈ {(1, 0, 0), (1, 1, 1)}. Now,

(P J )kB ∈ hF, P, Gi+, and Λ equals the “bold” isomorphism on hF, P, Gi+, with

range SO↑2,1Z. Thus w is the image either of (1, 0, 0) or of (1, 1, 1) under some element of SO↑2,1Z, a statement equivalent to (3) by Remark5.2.

(3) ⇒ (4) This is clear, since h(0, −1, 1), (0, 1, 1)i = −2 and h(1, 0, 1), (0, 1, 1)i = −1. (4) ⇒ (1) If ht, t0i = −1, then w ∈ Z3by the definition of w in Lemma5.1; assume

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other leg and the hypotenuse both odd. The condition t1t01+ t2t02− t3t03= −2 forces

t1, t01 to be both even and t2, t02 both odd (or conversely). Since t3, t03 are surely

both odd, all the entries in Lt0× Lt must be even; thus w ∈ Z3.

The stated characterization of [t, t0] being even/odd is clear from the previous proof.

By Theorem5.3, w is in the hF , P , J i+-orbit of one of (1, 0, 0), (1, 1, 1), (0, 1, 0),

(−1, 1, 1). Hence Rw is a conjugate either of R(1,0,0) = J , or of R(1,1,1) = P , or

of R(0,1,0)= F , or of R(−1,1,1)= J P J by a matrix in hF , P , J i+; in any case, it

belongs to hF , P , J i.

Finally, letS be the matrix in (13). By direct computation S = (θ − θ0)−1−θ − θ0 2θθ0

−2 θ + θ0 

,

which has the form α ββ ¯¯α, as can easily be checked; hence S ∈ PSU ±

1,1C. If we

can prove thatS has entries in Z[i], then necessarily S = Rw. Indeed, the matrix

S−1R

w would then belong to the fuchsian group PSU1,1Z[i], and would fix the two cusps θ, θ0; hence, it must be the identity matrix.

Write uniquely θ = κµ/¯µ, θ0 = λν/¯ν, as explained in §3. By Theorem 5.3, there existsA ∈ hF , P, J i+= PSU1,1Z[i] such that

A κµ λνµ¯ ν¯  ∈−i i 1 1  ,1 i 1 1  ,1 −1 1 1  , i −1 1 1  . This implies that the determinant δ = κµ¯ν − λ¯µν divides 2 in Z[i]. Since

θ θ0 1 1  =κµ λν ¯ µ ν¯   ¯µ ¯ ν −1 , we have θ θ0 1 1  Jθ θ 0 1 1 −1 =κµ λν ¯ µ ν¯  Jκµ λν ¯ µ ν¯ −1 = δ−1−κµ¯ν − λ¯µν 2κλµν −2¯µ¯ν λ¯µν + κµ¯ν  = δ−1δ − 2κµ¯ν 2κλµν −2¯µ¯ν δ + 2λ¯µν  , which has entries in Z[i].

6. Billiard maps. Having arranged our tools in working order, we proceed to our core objects.

Definition 6.1. A unimodular partition of the unit circle S1is a counterclockwise

cyclically ordered m-uple t0, t1, . . . , tm−1 of pythagorean triples, of cardinality at

least 3, such that each interval [ta, ta+1] is unimodular (including [tm−1, t0]; here

and in the following we are writing indices modulo m). We will write wa = (Lta+1×

Lta)/hta+1, tai ∈ S for the points defined by Lemma 5.1.

According to our conventions, and without further notice, we will often switch to a complex-numbers setting, thus writing θa for ta.

For every a, let labe the geodesic in D of ideal endpoints θa and θa+1; of the two

halfplanes determined by la, let Da be the one containing all other lb, for b 6= a.

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vertices θ0, . . . , θm−1, on which we can play billiards in the usual way. Namely, any

unit velocity vector attached to an infinitesimal ball in the interior of D determines an oriented geodesic g starting from an ideal point ρ and ending at σ. The ball travels along g at unit speed, until it hits the side la determined by the half-open

interval [θa, θa+1) to which σ belongs (unless σ is one of the vertices, in which

case the ball is lost at infinity). When hitting la, the ball rebounces with angle

of reflection equal to the angle of incidence, and continues its trajectory along the geodesic g0 which is the image of g with respect to the reflection with mirror la.

This reflection is induced by the matrix Rwa in (13) (with θ = θa and θ 0= θ

a+1),

and thus has ideal initial and terminal points Rwa ∗ ρ andRwa∗ σ, respectively. All of this naturally suggests the following standard definition [18, Chapter 6], [16, §IV.1].

Definition 6.2. The billiard map determined by the unimodular partition θ0, . . . ,

θm−1 is the map eB from (S1 × S1) \ (diagonal) to itself defined by eB(σ, ρ) =

(Aa∗ σ,Aa∗ ρ), where a is the index of the unique half-open interval Ia= [θa, θa+1)

containing σ, andAa=Rwa. The map eB is continuous, and determines a topolog-ical dynamtopolog-ical system. We denote by (S1, B) the factor system naturally induced

by the projection (σ, ρ) 7→ σ; in short, B(σ) =Aa∗ σ for σ ∈ Ia.

We will freely use Theorem 4.1 to conjugate eB to a map acting on any of the spaces (S1)–(S6); we will still denote the conjugated map by eB, slightly abusing notation. For ease of visualization (and crucially in §9 and §10) we will also con-jugate eB and B to maps on [0, 1)2\ (diagonal) and [0, 1), respectively; these last

conjugations are realized through the normalized (i.e., the image is divided by 2π) argument function arg : ∂D → [0, 1).

Example 6.3. The ordered 6-uple θ0= 1, θ1= 12 + 5i 13 , θ2= 4 + 3i 5 , θ3= i, θ4= −i, θ5= 4 − 3i 5 ,

is a unimodular partition, whose corresponding billiard table is shown in Figure 3

(left). The matricesA0, . . . ,A5 are

 −5i −1 + 5i −1 − 5i 5i  ,  −8i −4 + 7i −4 − 7i 8i  ,  −2i −2 + i −2 − i 2i  ,  −i i  =J ,  −2i 2 + i 2 − i 2i  ,  −3i 1 + 3i 1 − 3i 3i  . The graph of the arg-conjugate of B is shown in Figure 3 (right); it requires caution in two respects. First, B is a continuous map on S1 and, second, it is

piecewise-defined via six pieces, whose endpoints are given by the six B-fixed points (0 = 1 included). We plot in Figure4(left) 5000 points of the eB-orbit of a “typical” point in the de Sitter space S, and in Figure4(right) their arg-images. The cluster points apparent in this latter figure correspond to the six fixed points cited above. These are indifferent fixed points (i.e., the derivative of B has absolute value 1), and this forces the unique B-invariant measure absolutely continuous with respect to the Lebesgue measure to be infinite; see Theorem7.2and Figure5. Note that eB is not injective: the points (θ0,A0∗ θ2) and (A2∗ θ0, θ2) are different, but both get

mapped to (θ0, θ2) (see however Theorem7.1(i)).

We let ΓB±be the group generated byA0, . . . ,Am−1, and ΓB= ΓB±∩ PSU1,1Z[i] the associated fuchsian group. By conjugating with an appropriate element of

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θ0 θ1 θ2 θ3 θ4 θ5 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Figure 3. A unimodular billiard table and its associated factor map B. 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Figure 4. A typical eB-orbit on the de Sitter space and its arg-image PSU1,1Z[i] we always assume, without loss of generality, that θ0 = 1. As noted

in §2, ΓB± admits the presentation hx0, . . . , xm−1 | x20 = x21 = · · · = x2m−1 = 1i,

and hence is isomorphic to the free product of m copies of the group of order two. Equivalently stated, each element of ΓB± can be uniquely written as a word in the generators A0, . . . ,Am−1, subject to the only condition that the same generator

does not appear in two consecutive positions. Since D has finite hyperbolic area, ΓB and ΓB± have finite index in PSU

± 1,1Z[i].

Definition 6.4. Let B, I0, . . . , Im−1be as in Definition6.2. For each t = 0, 1, 2, . . .,

let atbe determined by Bt(σ) ∈ Iat; the point ϕ(σ) = a0a1a2. . . = a in the Cantor space {0, . . . , m − 1}ω is the B-symbolic sequence of σ.

Lemma 6.5. The B-symbolic-sequence map ϕ : S1→ {0, . . . , m − 1}ω is injective.

Its range is the set of all sequences a such that:

(i) if at= at+1for some t, then at= at+h for every h ≥ 0;

(ii) for any a ∈ {0, . . . , m − 1}, the tail of a is neither of the form a(a + 1), nor of the form (a − 1)a (the bar denoting periodicity).

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Remark 6.6. Since we are considering half-open intervals, each σ has precisely one B-symbolic sequence; thus ϕ is well defined. This differs slightly form other treatments of Gauss-like maps (see, e.g., [29, §2.1] or [45, §1.2.1]), in which rational points have two symbolic sequences. Note that ϕ is not continuous; indeed, if it were it would have compact image, which is not the case (e.g., all sequences of the form (01)n0 lie in the image, but the resulting sequence of sequences does not have

a limit point in ϕ[S1]).

Proof of Lemma 6.5. EachAais an involution, and exchanges IawithSb6=aIb, the

bar denoting topological closure. However, in this proof we carefully distinguish B (which maps bijectively Ia to Sb6=bIb) from Aa (which is one of the branches of

B−1, the one that maps bijectivelyS

b6=aIbto Ia). We do so in order to prepare the

ground for the proof of Theorem9.2, where the argument we are going to provide will be adapted to another (m − 1)-to-1 covering map of S1.

Let a = ϕ(σ). If at = at+1 = a, then Bt(σ) ∈ Ia ∩ B−1[Ia] = {θa}. Since

θa is a B-fixed point, we have at+h = a for every h ≥ 0. Moreover, if t ≥ 1 and

at−16= a, then we have θa= Bt(σ) ∈ B[Iat−1], which implies at−16= a − 1, because θa ∈ B[I/ a−1]. Hence a cannot have tail (a − 1)a. The fact that a cannot have

tail a(a + 1) is proved in [12, Theorem 2.1]. We conclude that every B-symbolic sequence must satisfy (i) and (ii).

Conversely, we fix a satisfying (i) and (ii) and show that there exists a unique point having a as B-symbolic sequence. We need a preliminary remark: suppose we know that σ is the unique point having B-symbolic sequence b. Then, by direct inspection, we have:

(a) if σ is in the interior of Ib0 and b 6= b0, thenAb∗ σ is in the interior of Ib and is the unique point having B-symbolic sequence bb;

(b) the same conclusion holds if σ = θb0, provided that b /∈ {b0, b0− 1}.

Case 1. The sequence a has tail a, say from time t on. If t = 0, then there exists a unique point having B-symbolic sequence a, namely θa. If t > 0, then the

previous remark and induction show thatAa0· · ·Aat−1∗ θa is the only point having B-symbolic sequence a.

Case 2. The sequence a does not have tail a, for any a. Since at6= at+1for every t,

we have strict inclusions Iat ⊃Aat[Iat+1] for every t, and hence a strictly decreasing sequence of nested intervals

Ia0 ⊃Aa0[Ia1] ⊃Aa0Aa1[Ia2] ⊃ · · · . (14) We claim that this sequence shrinks to a singleton. Indeed, each set in (14) is a unimodular interval, strictly containing the following one. By Lemma 5.1(v) the third coordinates of the corresponding points wa0, Aa0wa1, Aa0Aa1wa2, . . . on the de Sitter space form a strictly increasing sequence. Since we are dealing with unimodular intervals, these third coordinates are integer numbers, and a strictly increasing sequence of integers must go to infinity. Therefore the arclengths of the intervals go to 0, and the intersection of the sequence in (14) contains at least one point —by compactness— but no more than one.

Let σ be the shrinking point of (14) and let ϕ(σ) = b; we prove a = b by induction (note that, clearly, no point other than σ may have B-symbolic sequence a). We have σ ∈ Ia0∩ Ib0; if a0 were different from b0, then necessarily σ = θb0 and b0= a0+1. Therefore, for every t ≥ 1 we have σ = Bt(σ) ∈ Bt[Aa0· · ·Aat−1Iat] = Iat, and thus σ belongs to Iat. This implies a = a0(a0+ 1), which contradicts (ii);

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hence a0= b0. For the inductive step, assume ar= br for 0 ≤ r < t. Then Bt(σ)

has B-symbolic sequence btbt+1. . . and is the unique shrinking point of the chain

Iat ⊃Aat[Iat+1] ⊃AatAat+1[Iat+2] ⊃ · · · . Applying the base step above to Bt(σ) we get a

t= bt.

7. Natural extension and invariant measures. If ϕ(σ) has constant tail a for some a ∈ {0, . . . , m−1}, i.e., Bh(σ) = θafor some h, we say that σ is B-terminating.

If ϕ(σ) has periodic tail ah· · · ah+p−1with minimal preperiod h and period p ≥ 2,

we say that σ is B-periodic or B-preperiodic, according whether h is 0 or greater than 0.

We will push the identification of the de Sitter space with (S1× S1) \ (diagonal)

a bit further by using the symbol S for both; this is unambiguous since writing w ∈ S or (σ, ρ) ∈ S clearly distinguishes the two uses. With this understanding, we denote by SB the set of all pairs (σ, ρ) such that:

(i) both σ and ρ are B-nonterminating; (ii) σ and ρ belong to different intervals.

For the map B of Example6.3, the orbit in Figure4is dense in SB.

Theorem 7.1. The following facts hold. (i) eB  SB is a bijection on SB.

(ii) If (σ, ρ) ∈ S is such that both σ and ρ are B-nonterminating, then eBt(σ, ρ) ∈ SB for some t ≥ 0.

(iii) Let ˜µ be the PSU±1,1C-invariant measure on (S1× S1) \ (diagonal) given by

Theorem 4.1. Then (SB, ˜µ, eB) is a measure-preserving system, and so is its

factor (S1, µ, B), where µ = π

∗µ is the pushforward measure induced by the˜

projection π(σ, ρ) = σ.

(iv) The invertible system (SB, ˜µ, eB) is the natural extension of (S1, µ, B).

Proof. (i) The fact that eB maps SB into itself is clear. Writing f for the involution

(σ, ρ) 7→ (ρ, σ) of SB, it is also clear that f ◦ eB ◦f = eB−1on SB. In terms of symbolic

sequences, all of this just amounts to eB : (a0a1. . . , b0b1. . .) 7→ (a1. . . , a0b0b1. . .)

and f ◦ eB ◦ f : (a0a1. . . , b0b1. . .) 7→ (b0a0a1. . . , b1. . .).

(ii) Let σ 6= ρ be both B-nonterminating. By Lemma 6.5 there exists t ≥ 0 such that Bt(σ) and Bt(ρ) belong to different intervals. By the definitions of eB and of

SB, we have eBt(σ, ρ) ∈ SB.

(iii) Any measurable M ⊆ SBis the disjoint union M = ˙S{Ma: a ∈ {0, . . . , m−1}},

where Ma = {(σ, ρ) ∈ M : ρ ∈ Ia}. Thus eB−1M = ˙SaBe−1Ma = ˙SaAa[Ma] and,

as ˜µ Aa[Ma] = ˜µ(Ma), we have ˜µ( eB−1M ) = ˜µ(M ).

(iv) The set {σ ∈ S1 : σ is B-terminating} is clearly B-invariant and has

µ-measure 0; modulo this nullset and its π-counterimage, we have the commuting square (SB, ˜µ) (SB, ˜µ) (S1, µ) (S1, µ) e B π π B

By the very definition of the natural extension [41, p. 22], the metric system (SB, ˜µ, eB) is the natural extension of its factor (S1, µ, B) if the supremum of the

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family of measurable partitions

{ eBt(fibers of π) : t ≥ 0}

is —modulo nullsets— the partition of SBin singletons. This condition amounts to

the request that if (σ, ρ) 6= (σ0, ρ0), then there exists t ≥ 0 such that π eB−t(σ, ρ) 6= π eB−t(σ0, ρ0). This request is clearly satisfied: if σ 6= σ0 we take t = 0, while if

σ = σ0 we take t = h + 1, there h is the least nonnegative integer such that Bt(ρ)

and Bt0) lie in different intervals.

As usual in the context of Gauss-like maps, once a model of the natural extension has been determined the computation of the (unique) absolutely continuous B-invariant measure is easy; we state the result for the arg-conjugates of eB and B. Theorem 7.2. Let X = {(arg σ, arg ρ) : (σ, ρ) ∈ SB} ⊂ [0, 1)2 and write —abusing

language— eB and B for arg ◦ eB ◦ arg−1 and arg ◦B ◦ arg−1, respectively. For a = 0, . . . , m − 1, let xa= arg θa, and let ha : [0, 1) → R≥0 be the function defined by

ha(x) =

π tan(π(x − xa))

− π

tan(π(x − xa+1))

on (xa, xa+1), and having value 0 elsewhere. Then the following facts hold.

(i) The unique (up to constants) eB-invariant measure on X absolutely continuous with respect to the Lebesgue measure is d˜µ = π2 sin(π(x − y))−2

dx dy. (ii) The unique (up to constants) B-invariant measure on [0, 1) absolutely

contin-uous with respect to the Lebesgue measure is dµ = P

aha dx.

(iii) Both systems (X, ˜µ, eB), ([0, 1), µ, B) are ergodic and conservative.

Proof. (i) This is just a change of variables, easily performed in two steps. Let F1, F2: R2→ R2 be defined by F1(x, y) = π(x − y), π(x + y) = (x0, y0), F2(x0, y0) =  cos(x0+ y0) 1 − sin(x0+ y0), cos(−x0+ y0) 1 − sin(−x0+ y0)  = (ω, α).

Then F2◦ F1 is a bijection from [0, 1)2\ {diagonal} to (P1R × P1R) \ {diagonal}; indeed, it amounts to the componentwise application of C−1◦ arg−1, with C the

Cayley matrix. This implies that the pushforward of the infinite invariant measure (ω − α)−2dω dα of Theorem 4.1 via arg ◦ C is (F2◦ F1)∗ (ω − α)−2dω dα. One

now computes F2∗  1 (ω − α)2dω dα  = 1/2 sin2(x0)dx 0dy0, F1∗  1/2 sin2(x0)dx 0dy0  = π 2

sin2(π(x − y))dx dy. (ii) Let x ∈ (xa, xa+1). Then ha(x) is the integral

Z xa 0 π2dy sin2(π(x − y))+ Z 1 xa+1 π2dy sin2(π(x − y))

of the invariant density in (i) along the fiber {x} × [0, xa] ∪ [xa+1, 1].

(iii) It is easy to check that B2 satisfies Thaler’s conditions [47, p. 69(1)–(4)]. This implies that B2 is ergodic and conservative; therefore so is B and its natural

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0 0.2 0.4 0.6 0.8 1 50 100 150 200 250 300

Figure 5. The invariant density for the map of Example6.3 We draw in Figure5the invariant densityP

aha for the map B of Example6.3.

We note that, in case m = 3, a direct geometric proof of Theorem7.2(ii) was given by Ko lodziej and Misiurewicz, using Ptolemy’s theorem on quadrilaterals inscribed in a circle [30], [34].

8. The Lagrange theorem. Our next result is a version of Serret’s theorem (two real numbers have the same tail in their continued fraction expansion precisely when they are PSL±2 Z-equivalent [24, §10.11], [39]) in modern language.

Theorem 8.1. The map B and the group ΓB± are orbit equivalent. More precisely, given σ, σ0 ∈ S1, there existsA ∈ Γ±

B such that σ0 =A ∗ σ if and only if there exist

h, k ≥ 0 such that Bh(σ) = Bk0). In particular, if σ belongs to Q(i) then it is

B-terminating, its orbit landing in the unique vertex of D which is ΓB-equivalent

to σ.

Proof. We begin proving the last assertion, for which the ∂K setting is expedi-ent. Let then s be a rational point, and let (w0)3, . . . , (wm−1)3 ∈ Z be the third

coordinates of the points w0, . . . wm−1 of Definition 6.1. We need a preliminary

step.

Claim. By conjugating B by an appropriate element of SO↑2,1Z, we may assume that (w0)3, . . . , (wm−1)3 are all greater than 0, with at most one exception that

may equal 0.

Proof of Claim. By Lemma 5.1(v), the greater is the arclength of Ia, the smaller

is (wa)3, with (wa)3= 0 corresponding to arclength π. This implies that no more

than one of the above third coordinates may be negative or 0. Say that (wa)3< 0.

If Ia is even, then by Theorem 5.3we may conjugate B by the matrix in SO↑2,1Z

that sends wa to (0, 1, 0), and we are through. If Ia is odd, than we conjugate by

the matrix that sends wa to (1, 1, 1); the image of Ia will then have arclength π/2.

One of the new third coordinates may now have value 0, but none may have value −1 or less, since value −1 already corresponds to an arclength of 3π/2, and the sum of the arclengths would exceed 2π.

Having proved our claim we perform, if needed, this preliminary conjugation, which does not affect the validity of our statement; renaming indices, we assume (w0)3≥ 0 and (w1)3, . . . , (wm−1)3> 0. If s is one of t0, . . . , tm−1, we are through.

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lifting s and s0 to their canonical representatives (i.e., to pythagorean triples), we have the identity in Z3

s0= Aas = s − 2

hwa, si

hwa, wai

wa. (15)

Now, hwa, wai = 1 since wa ∈ S, and hwa, si > 0 since s is in the interior of Ia.

This implies that the third coordinate of s0is strictly less than the third coordinate of s, unless a = 0 and (w0)3 = 0, in which case we have equality. But the third

coordinates of s and s0 are positive integers, and the exceptional case of equality is always preceded and followed by nonexceptional cases. Hence the process must stop, and this may happen only when the B-orbit of s lands in one of the interval endpoints t0, . . . , tm−1.

For the first assertion, the “if” implication is clear. Assume σ0 =A ∗ σ. If one of σ, σ0 is in Q(i) then so is the other, and by the first part of the proof both σ and σ0 land in one of θ0, . . . , θm−1. Since the vertices of D are ΓB±-inequivalent, they

must land in the same θa. Let then σ, σ0 ∈ Q(i) and ϕ(σ) = a. As noted in §/ 6,

A factors uniquely as A = Ab0. . .Abr−1, for certain b0, . . . , br−1∈ {0, . . . , m − 1}. Let 0 ≤ h ≤ r be minimum such that ah6= br−1−h. Then

σ0 =Ab0· · ·Abr−1∗ σ =Ab0· · ·Abr−1Aa0· · ·Aar−1∗ B r (σ) =Ab0· · ·Abr−1−hAah· · ·Aar−1 ∗ B r(σ) =Ab0· · ·Abr−1−h∗ B h(σ).

By (a) in the proof of Lemma6.5, ϕ(σ0) = b0. . . br−1−hahah+1. . ., and Br−h(σ0) =

Bh(σ).

The bijection between ∂D ∩ Q(i) and rational points in ∂K extends to higher degrees.

Lemma 8.2. Let s = [s1, s2, s3] ∈ ∂K correspond to σ = (s1+ s2i)/s3 ∈ ∂D

as usual, and let ω = C−1 ∗ σ = (µ ◦ υ)(s) ∈ P1

R. Then Q(s) = Q(ω) and [Q(ω) : Q] = [Q(i)(σ) : Q(i)]. If Q(ω)/Q is Galois totally real, then the Galois groups Gal(Q(ω)/Q) and Gal(Q(i)(σ)/Q(i)) are naturally isomorphic. In particu-lar, assume that σ is quadratic over Q(i) and let σ0 be its Galois conjugate. Then σ0∈ ∂D and ω0= C−1∗ σ0 is the Galois conjugate of ω with respect to the quadratic

extension Q(ω)/Q.

Proof. Since the stereographic projection through [0, 1, 1] is a rational map with rational coefficients, the identity Q(s) = Q(ω) holds (with the convention that Q(∞) = Q). All statements follow from elementary Galois theory, as soon as one realizes that Q(i, σ) = Q(i, s1/s3, s2/s3). In this identity the left-to-right

contain-ment is obvious, and the other one follows from s1/s3= (σ + σ−1)/2.

The question of the validity of Lagrange’s theorem (preperiodic points correspond to quadratic irrationals) for the Romik map is left open in [42, §5.1]. It can be settled in the affirmative by the result in [38]; see also [14] for this issue, and [13] for diophantine approximation aspects of the Romik map. Here we provide a different proof, valid not only for the Romik map but for all maps based on unimodular partitions. Note that our proof covers not only Lagrange’s, but Galois’s theorem [40, Chapter III]: periodic points correspond to reduced irrationals.

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Theorem 8.3. The point σ ∈ S1 is B-preperiodic if and only if it is quadratic over Q(i). If this is the case and a0. . . ah−1ah. . . ah+p−1 is the B-symbolic sequence of

σ (with p the minimal period and h the minimal preperiod, so that ah−16= ah+p−1),

then the B-symbolic sequence of the Galois conjugate σ0 is a0. . . ah−1ah+p−1. . . ah.

In particular, the preperiodic σ is periodic iff so is σ0 iff (σ, σ0) ∈ SB.

Proof. Let σ be B-preperiodic. Clearly, for everyA ∈ PSU±1,1Z[i], we have Q(i)(A ∗ σ) = Q(i)(σ); we can then assume that σ is B-periodic, with B-symbolic se-quence a0a1. . . ap−1. Let B = Aa0Aa1· · ·Aap−1. By looking at the decreasing sequence (14) in the proof of Lemma6.5, we obtain

\

n≥0

Bn[I

a0] = {σ}.

SinceB ∗ σ is also in the above intersection, it equals σ, and this yields a quadratic polynomial with coefficients in Q(i) and having σ as root. This polynomial is not the zero polynomial, as B is not the identity matrix, and is irreducible over Q(i) because σ is B-nonterminating and Theorem8.1applies.

Conversely, let σ ∈ S1

be quadratic over Q(i). By Lemma8.2the conjugate σ0 is in S1 as well. For t ≥ 0, let eBt(σ, σ0) = (σ

t, σt0), and let gtbe the oriented geodesic

of origin σ0t and endpoint σt. By Theorem 7.1 there exists h ≥ 0 such that, for

0 ≤ t < h, the points σtand σ0tbelong to the same interval (so that gtdoes not cut

the billiard table D), while gtcuts D for every t ≥ h. In particular, the B-symbolic

sequences of σ and σ0 agree up to time h − 1 included, and disagree at time h. Let ω = C−1∗ σh, ω0= C−1∗ σh0; since σtand σt0 are still conjugate in Q(i)(σ)/Q(i), by

Lemma 8.2ω and ω0 are conjugate in Q(ω)/Q. Let O = {ξ ∈ Q(ω) : ξ(Zω + Z) ⊆ Zω + Z} be the coefficient ring of the module Zω + Z [8, Chapter 2 §2.2]. Then O is an order in Q(ω) with fundamental unit ε > 1, and thus the matrix

H =ω ω 0 1 1  ε ε0  ω ω0 1 1 −1 (16) (where ε0 is the conjugate of ε) is in PSL±2 Z.

Now, hF, P, J i = C−1 PSU±1,1Z[i]C is an index-3 subgroup of PSL±

2 Z (see the

end of the proof of Theorem 2.4), and ΓB± is a finite-index subgroup of PSU±1,1Z[i] (see §6). Hence, replacing H with an appropriate power, we obtain a matrix Hl = CHlC−1 ∈ Γ±

B which induces on D either a hyperbolic translation of

axis gh (if detHl = 1), or a glide reflection, again of axis gh (if detHl 6= 1).

As noted in §6, Hl can be uniquely written as Hl = Ab0· · ·Abq−1 for certain b0, . . . , bq−1 ∈ {0, . . . , m − 1}. We claim that b0· · · bq−1 and bq−1· · · b0 are the

B-symbolic sequences of σh and σh0, respectively (q might be a proper multiple of the

minimal period p); this will conclude the proof of Theorem8.3. We must have b06= bq−1. Indeed, if not, thenHl would factor as

Hl= (A

b0· · ·Abt−1)(Abt· · ·Abt+k−1)(Abt−1· · ·Ab0),

for some k ≥ 2, with t = (q − k)/2 and bt 6= bt+k−1. Hence gh would be the

(Ab0· · ·Abt−1)-image of the geodesic stabilized by (Abt· · ·Abt+k−1), which has end-points in the two distinct intervals Ibt and Ibt+k−1. Since btand bt+k−1are different from bt−1, the endpoints of gh would both lie in Ib0, which is impossible since gh cuts D; therefore b06= bq−1.

The sequence b0· · · bq−1satisfies (i) in Lemma6.5(because b06= bq−1), as well as

(ii) (because otherwise Hlwould be a power of some A

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t0 t1 t2 t3 t4 t5

Figure 6. A periodic orbit in a billiard table

parabolic, which is not possible because any power of the matrix in (16) has trace of absolute value greater than 2). Therefore, b0· · · bq−1 is the B-symbolic sequence

of a unique point of S1, and this point is necessarily σ

h, because σh is the ideal

endpoint of gh, and thus the shrinking point of

\

n≥0

(Ab0· · ·Abq−1) n[I

b0].

The same argument, applied toH−1 =Abq−1· · ·Ab0, shows that σ 0

hhas B-symbolic

sequence bq−1· · · b0.

Example 8.4. Consider the unimodular partition given by the pythagorean triples

t0=   1 0 1  , t1=   3 4 5  , t2=   0 1 1  , t3=   −1 0 1  , t4=   −4 −3 5  , t5=   0 −1 1  ;

in Figure6we draw the corresponding billiard table by thick geodesics.

Let q(x, y) = 4091x2+ 1302xy + 101y2, which has discriminant D = 42440. The

roots of q(x, 1) are ω0= −1302 +√D 2 · 4091 ' −0.13395, α0= −1302 −√D 2 · 4091 ' −0.18430. We work directly on the de Sitter space; by (9), q corresponds to

1 √ D   1 −1 1 1 1     4091 −1302 101  ∈ S.

Since we may safely multiply by a constant, and we prefer working with integer vectors, we multiply by√D/2 and define

v =1 2   1 −1 1 1 1     4091 −1302 101  =   −651 −1995 2096  ∈ √ D 2 S ∩ Z 3.

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